"Calculus and its applications: " - Marginals and differentials have objective: find marginal cost, revenue, and profit, find ∆y and dy, Use differentials for approximations. | 2012 Pearson Education, Inc. All rights reserved Slide Marginals and Differentials OBJECTIVE Find marginal cost, revenue, and profit. Find ∆y and dy. Use differentials for approximations. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITIONS: Let C(x), R(x), and P(x) represent, respectively, the total cost, revenue, and profit from the production and sale of x items. The marginal cost at x, given by C (x), is the approximate cost of the (x + 1)th item: C (x) ≈ C(x + 1) – C(x), or C(x + 1) ≈ C(x) + C (x). Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Slide DEFINITIONS (concluded): The marginal revenue at x, given by R (x), is the approximate revenue from the (x + 1)th item: R (x) ≈ R(x + 1) – R(x), or R(x + 1) ≈ R(x) + R (x). The marginal profit at x, given by P (x), is the approximate profit from the (x + 1)th item: P (x) ≈ P(x + 1) – P(x), or P(x + 1) ≈ P(x) + P (x). Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Slide Example 1: Given find each of the following: a) Total profit, P(x). b) Total cost, revenue, and profit from the production and sale of 50 units of the product. c) The marginal cost, revenue, and profit when 50 units are produced and sold. Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Slide Example 1 (continued): a) b) Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Slide Example 1 (concluded): c) So, when 50 units have been made, the approximate cost of the 51st unit will be $6200, and the approximate revenue from the sale of the 51st unit will be $6340 for an approximate profit on the 51st unit of $140. Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Slide Example 2: For x = 4, and ∆x = , find ∆y. Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Slide | 2012 Pearson Education, Inc. All rights reserved Slide Marginals and Differentials OBJECTIVE Find marginal cost, revenue, and profit. Find ∆y and dy. Use differentials for approximations. 2012 Pearson Education, Inc. All rights reserved Slide DEFINITIONS: Let C(x), R(x), and P(x) represent, respectively, the total cost, revenue, and profit from the production and sale of x items. The marginal cost at x, given by C (x), is the approximate cost of the (x + 1)th item: C (x) ≈ C(x + 1) – C(x), or C(x + 1) ≈ C(x) + C (x). Marginals and Differentials 2012 Pearson Education, Inc. All rights reserved Slide DEFINITIONS (concluded): The marginal revenue at x, given by R (x), is the approximate revenue from the (x + 1)th item: R (x) ≈ R(x + 1) – R(x), or R(x + 1) ≈ R(x) + R (x). The marginal profit at x, given by P (x), is the approximate profit from the (x + 1)th item: P (x) ≈ P(x + 1) – P(x), or P(x + 1) ≈ P(x) + P (x). Marginals and Differentials .
